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	      \begin{table}[htbp] 
  \caption{\bf Correlations:  local-PCA vs IPCA }
\medskip
\setlength{\tabcolsep}{0pt}
\begin{tabular}{lccccccc|} 
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
& \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:} & (0.83 , 0.95) &    (0.55,0.94) &  (0.91,0.98) & (0.62,0.99) & (0.90,0.99) & (0.49,0.99)  \\ 
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.48 , 0.98) &  (0.39, 0.99) &  (0.54,0.99) & (0.23,0.99) & (0.50,0.99) & (0.14,0.99) \\  \hline
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:} & (0.83 , 0.48) & (0.55,0.43) &  (0.91,0.55) & (0.62,0.55) & (0.90,0.62) &  (0.49,0.58)  \\
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.48 , 0.27) &  (0.39,0.26) &   (0.54,0.26) &  (0.23,0.25) &  (0.50,0.27) & (0.14,0.26) \\   \hline 
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{(local-PCA ,IPCA) } & \scriptsize{ (local-PCA ,IPCA)  } & \scriptsize{ (local-PCA ,IPCA) }  \\ 
 \scriptsize{ correlation ${\bf F }$:} & (0.82 , 0.39) &  (0.81,0.37) & (0.89,0.37) & (0.86,0.37) &  (0.88,0.23) & ( 0.83,0.26) \\
 \scriptsize{ correlation ${\bf \Lambda }_{t-1} $:} & (0.54 , 0.27) &  (0.62,0.29)   &  (0.59,0.26) & (0.66,0.28) &  (0.56,0.23) & (0.51,0.26) \\  \hline
    \end{tabular}
    \parbox{7.25in}{\footnotesize{This  table presents the absolute value of the correlations (averaged across $1,000$ Monte Carlo iterations) between the true and estimated first risk factor  $ { F }_{1t}$ (first row of each panel) and between the true and estimated  first risk exposure  $ {\lambda }_{1it-1} $ (second row of each panel), for various combinations of $(N,T)$. The data generating process is (OA.38) when the $10$-dimensional vector of instruments ${\bf z }_{it-1}$ is a stationary VAR (Experiment (A))  and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the estimated model is misspecified (one instrument instead of $10$ instruments).
}}
\label{FIG15MCOA}
\end{table}



   \begin{table}[htbp] 
  \caption{\bf   Total-$R^2$:  local-PCA vs IPCA }
\medskip
\setlength{\tabcolsep}{0pt}
\begin{tabular}{lccccccc|} 
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
& \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA)} & \scriptsize{(local-PCA,IPCA)} & \scriptsize{(local-PCA,IPCA) }  \\ 
 \scriptsize{ total-$R^2$ (in-sample):} & (0.86,0.61) &    (0.51,0.59) &  (0.81,0.52) & (0.34,0.50) & (0.83,0.50) & (0.34,0.47)  \\ 
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.27,0.57) &  (0.27,0.57) &  (0.15,0.51) & (0.14,0.49) & (0.16,0.50) & (0.17,0.47)  \\  \hline
  & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
  & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{ (local-PCA,IPCA) }  
  \\ 
 \scriptsize{  total-$R^2$ (in-sample):} & (0.86,0.06) &  (0.51,0.05) & (0.81,0.05) & (0.34,0.05) &  (0.81,0.05)  & (0.34,0.05) \\
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.27,0.04) &  (0.27,0.04) &   (0.15,0.03) &  (0.14,0.03) &  (0.16,0.03) & (0.17,0.03) \\   \hline 
    & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{(local-PCA,IPCA) } & \scriptsize{ (local-PCA,IPCA)  } & \scriptsize{ (local-PCA,IPCA) }  
  \\ 
 \scriptsize{ total-$R^2$(in-sample):} & (0.89,0.31) &  (0.71,0.33) & (0.84,0.05) & (0.63,0.06) &  (0.84,0.02) &  (0.65,0.01)\\
 \scriptsize{ total-$R^2$ (out-of-sample):} & (0.44,0.21) &  (0.57,0.22)   &  (0.34,0.04) & (0.49,0.04) &  (0.36,0.01) & (0.50,0.01) \\  \hline
    \end{tabular}
    \parbox{7.25in}{\footnotesize{This  table presents the 
  total-$R^2$ metric of Kelly et al (2019)[Eq. (15)]  (averaged across $1,000$ Monte Carlo iterations) defined as $ R^2 \equiv 1 - \sum_{i=1}^N \sum_{s=1}^{T_0} ( x_{is} -  \boldsymbol{\tilde{ \lambda}}_{is-1} \tilde{\bf f}_s )^2/ \sum_{i=1}^N \sum_{s=1}^{T_0} x_{is}^2 $, where $\boldsymbol{\tilde{ \lambda}}_{is-1} ,\tilde{\bf f}_s  $ denote the  estimated risk exposures and factors by either local-PCA or IPCA, evaluated in-sample and out-of-sample,  for various combinations of $(N,T)$. The data generating process is (OA.38)
  when the $10$-dimensional vector of instruments ${\bf z }_{it-1}$ is a  stationary VAR (Experiment (A))  and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the estimated model is misspecified (one instrument instead of $10$ instruments).
}}
\label{FIG16MCOA}
\end{table}




 \begin{table}[htbp]  
   \caption{\bf Pricing Ability:  local-PCA vs  IPCA }
\medskip
\setlength{\tabcolsep}{0pt}
\begin{tabular}{lccccccc|} 
 $ \scriptsize{(N,T):}$ & (15,5)  & (15,25) &  (100,5) & (100,25) & (1,000,5) & (1,000,25) \\  \hline 
& \multicolumn{5}{c}{\scriptsize{\bf Experiment (A): Correct Model - Stationary Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\ 
 \scriptsize{
 In-Sample:} &    5.851   &  2.036 &   5.213 &    1.479  &  5.304 &   2.150 \\
 \scriptsize{ 
 Out-of-Sample:} &  1.137 &  1.009 &    1.454 &    1.487 &    1.223 &    2.130 \\  \hline
  & \multicolumn{5}{c}{\scriptsize{\bf Experiment (B): Misspecified Model - Stationary Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\ 
  \scriptsize{  
 In-Sample:}   &     11.668 &   3.891 &    6.997 &    2.764 &  10.023 &   3.204 \\
 \scriptsize{ 
 Out-of-Sample:} &     2.145  &   3.401 &    1.967 &    2..002 &   2.574 &    3.012 \\  \hline
   & \multicolumn{5}{c}{\scriptsize{\bf Experiment (C): Correct Model - Random Walk Instruments} } & \\
  & \scriptsize{ (IPCA/local-PCA) } & \scriptsize{ (IPCA/local-PCA)  } & \scriptsize{(IPCA/local-PCA) } & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA)} & \scriptsize{(IPCA/local-PCA) }  \\
 \scriptsize{  
 In-Sample:} &  9.980 &   4.568  &    9.025 &    3.356 &   11.655 &    5.924 \\
 \scriptsize{ 
 Out-of-Sample:} &    1.681 &    3.494 &   1.942 &    2.849 &   2.077 &    4.436 \\ \hline
    \end{tabular}
  \parbox{7.0in}{\footnotesize{This  table presents ratios of the pricing ability criterion $N^{-1}\sum_{i=1}^N \tilde{\delta}_{i}^2  $  corresponding to the IPCA  (numerator) and  local-PCA (denominator),   averaged across $1,000$ Monte Carlo iterations,  and evaluated in-sample and out-of-sample,  for various combinations of $(N,T)$. The data generating process is (15) where the  vector ${\bf z }_{it-1}$ is a $10$-sized stationary VAR (Experiment (A)) and a multivariate random walk (Experiment (C)), respectively. Experiment (B) differs from (A) because the IPCA-estimated model is misspecified (one instrument instead of the $10$ required instruments).}}
\label{Table1OA}
\end{table}

	 

	
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